Not every set of classical variables lead to valid quantum operators, neither for CQ or for AQ.The
correct choice of classical variables to promote to quantum operators for CQ are Cartesian coordinates, which creat a constant vanishing curvature. The classical coordinates that creat a constant negative curvatureA constant negative curvature is discussed in [1]}. are the correct coordinates to promote to quantum operators for AQ.Classical variables for spin variables that form a constant positive curvature are the final
set of correct classical variables that promote to valid quantum operators. The constant (+, 0, -1)
curvatures are all addressed in [2]}, Sec. 1.
Regarding CQ and AQ operators, the principal difference is that CQ operators must span the whole real line, while the AQ operator \(Q\)  strictly spans all positive coordinates, all negative coordinates, or both, provided \(Q\ne 0\) .Instead of \(-\infty <Q\ne 0<\infty \) , we can also have \(-\infty <Q\ne b<\infty \) , provided \(b\)  is finite.
